Is it true to say that for every non trivial A,B in P (or even in R) it holds that A is mapping reducible to B? (A <m B)

Since there exists MA that decides A in poly time and since B is non trivial then there exists w1 in B and w2 not in B and
therefore we could define a computable function f that on input x emulates MA on x and returns w1 if MA accepts and w2 otherwise.

No. If B is nontrivial there exists w1 in B and w2 not in B and we do not "waste time" finding them - we can simply use them.
This is because the language B is given to us in advance, and not part of the input.